A generalized delay-induced SIRS epidemic model with relapse

In this paper, a generalized delay-induced $SIRS$ epidemic model with nonlinear incidence rate, latency and relapse is proposed. Our epidemic model is a generalized one, and the published epidemic models are the special cases of ours under some conditions. By using LaSalle's invariance principle and Lyapunovi's direct method, the dynamical behaviors are investigated and the results show that the disease free-equilibrium $Q_0$ is globally asymptotically stable if the basic reproduction number $R_0<1$ for any time delay. However, if the basic reproduction number $R_0>1$, there exists a unique endemic equilibrium $Q_{\ast }$ which is locally asymptotically stable under some conditions. Moreover, the effects of latency and relapse on the transmission dynamics of the diseases are analyzed by some numerical experiments which conducted based on $ODE45$ in Matlab.